The dual complex of log Calabi-Yau pairs on Mori fibre spaces

TL;DR

This paper investigates the structure of the dual complex of certain log Calabi-Yau pairs on Mori fibre spaces, showing it is a finite quotient of a sphere under specific conditions, advancing understanding in algebraic geometry.

Contribution

It proves that the dual complex of a dlt log Calabi-Yau pair on a Mori fibre space is a finite quotient of a sphere when the Picard number or the base dimension is at most 2.

Findings

Dual complex is a finite quotient of a sphere under given conditions.

Provides partial answer to Kollár and Xu's question on dual complexes.

Extends understanding of dual complexes in algebraic geometry.

Abstract

In this paper we show that the dual complex of a dlt log Calabi-Yau pair $(Y, \Delta)$ on a Mori fibre space $\pi: Y \to Z$ is a finite quotient of a sphere, provided that either the Picard number of $Y$ or the dimension of $Z$ is $\leq 2$. This is a partial answer to Question 4 in "The dual complex of Calabi-Yau pairs" by Koll\'ar and Xu.